In the next theorem, we collect some results about the spectrum of \(\mathsf {RCD}(K,\infty )\)  spaces which will be useful later on. For the compact embedding \(W^{1,2}(X,\mathsf {d},\mathfrak {m}) \subset \subset L^2(X,\mathfrak {m})\)  in \(\mathsf {RCD}(K,\infty )\)  spaces under different assumptions see [1]}.
